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%           FPSAC/SFCA '94, May 23-27, 1994   at DIMACS
%                     Problem Session
%   LaTeX file of problem submitted by Jim Propp (May 27, 1994)
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\begin{center}
{FPSAC/SFCA '94}\\
{PROBLEM SESSION}\\
{Submission received May 27, 1994}\\[.2in]
{\bf Jim Propp}\\[.2in]
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\noindent
Fix $1 \leq k \leq n$.
Define 0-1 random variables $X_1,X_2,...,X_n$
such that for all $n$-tuples $(x_1,x_2,...,x_n) \in \{0,1\}^n$,
$\Prob[X_i=x_i \ \mbox{for all $i$}] = 0$
unless $\sum_{i=1}^n x_i = k$,
in which case
\[
\Prob[X_i = x_i \ \mbox{for all $i$}] =
\frac{1}{2^{k(n-k)}}
\left( \prod_{1 \leq i < j \leq k} \frac{a_j-a_i}{j-i} \right)
\left( \prod_{1 \leq i < j \leq n-k} \frac{b_j-b_i}{j-i} \right) \ ,
\]
where $\{a_1,a_2,...,a_k\} = \{i:\ x_i = 1\}$
and $\{b_1,b_2,...,b_{n-k}\} = \{i:\ x_i = 0\}$
($a_1<a_2<...<a_k$,\ $b_1<b_2<...<b_{n-k}$).
This is the distribution on zig-zag patterns in the $k$th diagonal
of the Aztec diamond, induced by a domino tiling chosen uniformly
at random.

\noindent
{\bf Question:}
Are the $X_i$'s (nonstrictly) negatively correlated?

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